determine for what numbers, if any, the given function is discontinuous. construct a table to find any…

Explanation:

Step1: Recall continuity condition

A function $f(x)$ is continuous at $x = a$ if $\lim_{x\rightarrow a}f(x)=f(a)$. We need to check the limit as $x\rightarrow0$ and compare with $f(0)$.

Step2: Find the limit as $x\rightarrow0$

We use the well - known limit $\lim_{u\rightarrow0}\frac{\sin u}{u} = 1$. Let $u = 20x$, as $x\rightarrow0$, $u\rightarrow0$. Then $\lim_{x\rightarrow0}\frac{\sin20x}{x}=\lim_{x\rightarrow0}\frac{\sin20x}{x}\cdot\frac{20}{20}=20\lim_{x\rightarrow0}\frac{\sin20x}{20x}$. Since $\lim_{u\rightarrow0}\frac{\sin u}{u} = 1$ with $u = 20x$, we have $20\lim_{x\rightarrow0}\frac{\sin20x}{20x}=20\times1 = 20$.

Step3: Compare with $f(0)$

We are given that $f(0)=20$, and $\lim_{x\rightarrow0}f(x) = 20$. For all non - zero $x$, the function $\frac{\sin20x}{x}$ is a well - defined rational function (with $\sin20x$ and $x$ being well - defined).

Answer:

B. The function is continuous everywhere.